Answer
\[ - \frac{{{{\cos }^3}x}}{3} + \frac{{{{\cos }^5}x}}{5} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {{{\cos }^2}x{{\sin }^3}xdx} \hfill \\
\hfill \\
split\,\,{\sin ^3}x \hfill \\
\hfill \\
\int_{}^{} {{{\cos }^2}x{{\sin }^2}x\sin x\,dx} \hfill \\
\hfill \\
substitute\,\,{\sin ^2}x = 1 - {\cos ^2}x \hfill \\
\hfill \\
\int_{}^{} {{{\cos }^2}x\,\left( {1 - {{\cos }^2}x} \right)\sin xdx} \hfill \\
\hfill \\
Distribute \hfill \\
\hfill \\
\int_{}^{} {\,\left( {{{\cos }^2}x\sin x - {{\cos }^4}x\sin x} \right)dx} \hfill \\
\hfill \\
rewrite \hfill \\
\hfill \\
- \int_{}^{} {{{\cos }^2}x\,\left( { - \sin x} \right)dx + \int_{}^{} {{{\cos }^4}x\,\left( { - \sin x} \right)dx} } \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
- \frac{{{{\cos }^3}x}}{3} + \frac{{{{\cos }^5}x}}{5} + C \hfill \\
\hfill \\
\end{gathered} \]
